Friction and Wear of Materials Project: Comparison of a Ball bearing with Steel balls vs Ceramic balls Brady Walker 11/19/07

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Friction and Wear of Materials


Project: Comparison of a Ball bearing with Steel balls vs Ceramic balls



Brady Walker

11/19
/07



Table of Contents


List of Symbols











iii

List of Tables












iv

List of Figures












iv

Abstract












1

1.

Introduction











2

2.

Description of

Ball Bearing Surfaces in Contact






4

3.

Development of Theory for Elastic Contact and Subsurface Stress




15

4.

Friction & Wear, Contact Fatigue (9.4 in reference)






30

5.

Lubrication (EHD)










40

6.

Conclusion











50

References












51


Abstract


This paper discusses the Tribological differences between ball bearings manufactured with steel and ceramic
balls. Though the topic of ball bearing design is mature in the bearing industry, new materials
have been
presenting themselves as
viable options. As with any material selection there are “trade
-
offs”, determining
these “trade
-
offs” is the focus of this paper. The evaluation of two materials will be divided into 4 major
topics. The first topic is
a discussion of the contact surface topography, the next is a discussion of the Hertz
theory and the variation in point contact stresses (both surface and subsurface), the next topic is a grouping of
friction/wear/contact fatigue, and finally the last topi
c is lubrication (elastohydrodynamic).


1.0

Introduction


Ball bearings are devices that allow relative motion between a rotating object and a stationary object with low
friction. Ball bearings contain 4 essential components; an inner ring, outer ring, rolli
ng elements and usually a
cage (see figure #1 for a schematic of a typical ball bearing). Typically, there are thrust and/or radial loads
applied to bearing as it rotates. If the speed of the bearing is sufficient, centrifugal loads are generated (due to

the mass of the balls) on the outer raceway. For high speed applications, there has been an interest to use
ceramic (silicon nitride) balls to lessen the effect of centrifugal loading. Silicon Nitride has a density about
40% than that of steel. Beari
ng designs are largely dependant upon the maximum hertzian contact stress
developed on the raceways. Both surface and subsurface hertzian stresses are of interest.




Figure #1 Sample Ball Bearing



Outer Ring

Inner Ring

Cage

Balls


2.0

Description of Ball Bearing Surfaces in
Cont
act

In a traditional ball bearing there are several surfaces that are of importance
, the following diagram defines
typical surface roughness although the only surfaces that we will be discussing are the ball and raceway
roughness.


Bearing raceways

and balls are ground, honed and lapped. The following figure is a sample
topography of contacting surfaces.


Ball Roughness:

Steel = 1.0 AA

Ceramic = 0.1 AA

Raceway Roughness= 3.0 AA

3.0

Development of Hertzian Contact Theory


The contact zone between the rolling element and the raceway (inner or outer) creates an ellipsoida
l surface
compressive stress distribution, see figure no 2. The maximum compressive hertzian contact stress is given as:

ellipse
contact

projected

of

axis
semiminor

b
ellipse
contact

projected

of

axis
semimajor

a
load

,
2
3
max




ball
Q
where
ab
Q



The elliptical contact area is shown in figure number 2, the semimajor and semiminor axes are also
shown.



Figure No. xxx General Form of Two Point Contact





Figure No. xxx Ball Bearing Two Point Contact



z

z

x

(1)

y


Figure 2: Two bodies in point contact, resulting elliptical contact area.



The normal stress at other points within the contact area is given
by the following equation:

distances
direction

principle

are
y

and
x
,
1
2
3
2
1
2
2
where
b
y
a
x
ab
Q




























The values of a and b are determined from the following equations:

(2)

kind
first

of

integral

elliptical

complete

kind

second

of

integral

elliptical

complete

2
2
2
*
2
a
:
and

bodies

contacting
in

points

remote

of
approach

relative

the
is


where,
II
body
for

elasticity

of

Modulus

E
I
body
for

elasticity

of

Modulus

E
II
body
for

ratio

poissons

I
body
for

ratio

poissons

2
1
1
2
3
,
1
1
2
3
1
1
2
3
2
*
3
1
3
1
2
*
II
I
II
3
2
2
2
2
*
3
1
2
2
2
*
3
1
2
2
2
*


























































































































b
a
k
k
k
b
k
E
E
Q
ly
additional
E
E
Q
b
b
and
E
E
Q
a
a
I
II
I
I
II
I
I
II
I
I



diameter
pitch

d
angle
contact

Dcos

diameter

ball

D
curvature

ring
outer

curvature

ring
inner

ring
outer
for


1
2
1
4
1
ring
inner
for


1
2
1
4
1
,
m









































m
o
i
o
i
d
f
f
f
D
f
D
also


Hertz’s analysis is applied only to surface stresses caused by a concentrated force ap
plied perpendicular to the
surface. Experimental evidence shows that failure of rolling bearings in surface fatigue caused by this load
emanates from points below the stressed surface. The depths at which significant stresses occur below the
surface are
of interest




Example 1


A ball bearing containing a 1” diameter ball, is contacting the inner raceway of a ball bearing with a 52%
curvature. The ball load is 500 lbf, calculate the following:

1.

the width of the semimajor and semiminor contact a
rea

2.

maximum hertzian contact stress

3.

maximum shear stress

4.

depth to max shear stress


1.0

Raceway curvature = raceway
radius/ball diameter=.52









09969
.
0
402
.
2
500
738
.
3
0045
.
0
finally,
A
appendix
in

table
from

ted
(approxima

738
.
3
*
402
.
2
1397
.
1
1397
.
2
52
.
1
4
1
1
1397
.
2
.
6
30
cos
0
.
1
,
"
2
.
6
30
cos
52
.
0
"
1
D

1
2
1
4
1
500
,
*
0045
.
0
3
1
3
1





















































a
a
theref ore
d
d
D
f
f
D
lbf
Q
where
Q
a
a
m
m
i
i



















0111
.
0
402
.
2
500
4166
.
0
0045
.
0
finally,
A
appendix
in

table
from

ted
(approxima

4166
.
0
*
402
.
2
1397
.
1
1397
.
2
52
.
1
4
1
1
1397
.
2
.
6
30
cos
0
.
1
,
"
2
.
6
30
cos
52
.
0
"
1
D

1
2
1
4
1
500
,
*
0045
.
0
3
1
3
1





















































b
b
therefore
d
d
D
f
f
D
lbf
Q
where
Q
b
a
m
m
i
i












2.0









psi
ab
Q
Q
549
,
215
01111
.
0
09969
.
0
2
500
3
2
3
max








3.0

















psi
S
S
finally
psi
S
S
psi
S
y
z
xy
z
z
y
330
,
129
664
,
64
193994
2
1
2
1
,
994
,
193
549
,
215
9
.
0
9
.
0
9
.
0
665
,
64
549
,
215
3
.
3
.
S
so,
B
appendix
in

figure

from

3
.
max
max
max
y
max

























4.0





"
005
.
0
01111
.
467
.
467
.



b
z




Figure No. xxx Stee
l
-
Steel, Ceramic
-
Steel Point Contact Stress Fields



Figure No. xxx Steel
-
Steel, Ceramic
-
Steel semimajor (a) contact ellipse shape


Max Hertzian
Contact
Stress (psi)

Cer
amic

Steel

3.0

Friction, Wear & Contact Fatigue


Rolling element bearings are sometimes called “antifriction” bearings to emphasize the

small amount of
frictional power consumed during their operation. Some friction does occur from other components within the
bearing such as seals, and cages which are much larger than the rolling friction. Bearing friction is manifested
as temperature r
ise in the bearing structure and lubricant unless proper heat removal methods are used. When
excessive temperature level occurs, the rolling bearing steel suffers loss in its ability to resist rolling contact
fatigue and the lubricant becomes ineffective.

The result is rapid bearing failure.


Rolling friction is the resistance to motion that takes place when a surface is rolled over another surface. With
hard materials the coefficient of rolling friction between a spherical body against itself or a flat
body generally
is in the range of 5x10^(
-
3) to 10^(
-
5). In comparison the coefficient of sliding friction of dry bodies ranges
typically from 0.1 to 1.0.


From the standpoint of comparing steel vs ceramic rolling elements. The friction coefficients are a
s follows:



4.0

Lubrication (EHD)